DCA6108 DISCRETE MATHEMATICS & GRAPH THEORY

Description

Discrete Mathematics & Graph Theory

 

 

 

1. Find the inverse of the matrix A = [[1,2,3],[0,4,5],[1,0,6]] using the adjoint method. (10 Marks)

Ans 1.

Introduction

Another matrix that can be multiplied with a square matrix to produce the identity matrix is called its inverse matrix. Invertibility of a matrix is related to the determinant of the matrix as not all matrices have inverses. One of the standard methods for calculating the inverse of a matrix is the adjoint method, which is applicable to small square matrices, e.g., a three by three matrix. This approach is based on cofactors and the determinant of matrix.

Meaning of the Adjoint Method

The adjoint method is to use the relationship that the inverse of a matrix is equal to the adjoint of the matrix divided by the

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2. Find the solution of the following system using gauss elimination: 3x + 4y + z = 10, 2x – 3y + 5z = -9, x + 2y – z = 6. (10 Marks)

Ans 2.

Introduction

There are many different ways to solve a system of linear equations with multiple unknowns; one of the most reliable methods is the “Gauss Elimination”. This is a way of transforming the given system to an upper triangular form by using a series of row operations. After reducing the system to this simplified form, the values of the unknowns can be determined step by step by back substitution.

Idea used behind the Gauss Elimination

The concept behind Gauss elimination is that it eliminates one term from each of the equations below the pivot equation. This is accomplished by adding and/or subtracting appropriate multiples of one equation to another until a specific variable is eliminated. This is continued until the system has the form of an upper triangular matrix, with the last equation having only one

 

 

3. Let f:A -> B, g:B -> C, and h:C -> D. If f(x) = x + 4, g(y) = 3y and h(z) = z – 1, find the composition h o g o f(x), also check the bijective ness of mappings. (10 Marks)

Ans 3.

Introduction

Compositions of functions are functions that are applied in succession. The concept is very prevalent in mathematics, where the result of a function is used as the input to another function. In addition to identifying the composition, it is important to see if each mapping is bijective, that is, one to one and onto simultaneously.

Meaning

 

 

4. Check this (p -> q) v (not q -> not p) = True or not. (10 Marks)

Ans 4.

Introduction

In logic, a statement that is always true no matter what truth values its individual parts take is called a tautology. A truth table is created for all possible combinations of truth values of the statements in the logical expression to determine if it is a tautology. If the expression is true in every row of the truth table, it is true and confirmed as a tautology.

The conditional and contrapositive

 

 

 

5. The table shows the height distribution (in cm) of students in a school: 140-150: 5, 150-160: 10, 160-170: 12, 170-180: 8, 180-190: 5. Find the median height of the students. (10 Marks)

Ans 5.

Introduction

In grouped data (where the data is given in intervals and the number of items in each interval), the most common approach for locating the median value can no longer be applied. However, the median must be calculated using a special formula for continuous grouped frequency distributions. This formula is to find the class where the median actually belongs and is based on the cumulative frequency.

For a grouped

 

 

6. Explain the degree of a vertex in an undirected graph and binary trees. (10 Marks)

Ans 6.

Introduction

In discrete mathematics, an important class of structures are graphs and trees, which are used to model relationships among various objects. The degree of a vertex is one of the fundamental characteristics that are evaluated in a graph and it is a measure of the number of connections that a given vertex has with other vertices. There is another type of structure based on